Challenges cover a variety of topics
This is the final Captivating Circles challenge, this challenge requires the concepts of geometry and trigonometry.
We begin this challenge with 2 concentric circles. The smaller circle divides the larger 1 into 2 equal pieces. We then draw 2 parallel chords in the large circle, which divide it evenly into three. They do not divide the small circle into three equal pieces. The challenge is then to find the area of the biggest piece in the smallest circle if the large circle has radius 1.
Today, we have a new captivating circle, and it is the first challenge to require the use of combinatorics. It also requires some trig and pathmaking.
Our second captivating circle is a circle with 7 points on its circumference. The points are equally spaced. Each point connects to 2 others through the circle and 2 others through a straight line. No points can connect to another through both a line and arc. The circle has circumference π. How many possible different sets of connections can be made? And of these, what is the highest possible distance that must be traversed to get between 2 points?
Today we begin a special series, captivating circles. This challenge requires trigonometry and geometry.
The first captivating circles challenge is to find the area of the shaded region in the circle in the diagram. All vertices of the white triangle are on the circumference of the big circle, and the small circle is tangent to all 3 sides of the triangle. You may find that this problem requires theorems and formula’s not normally required. Do not be afraid to research the tools you need to solve this problem or other captivating circles, it’s knowing which tools to use that’s the challenge.
Solving this challenge requires an understanding of geometry, trigonometry, and angle theorems.
In today’s challenge, we face two triangles who are not congruent. This is surprising because they appear identical, and we are told they share two sides. We can find that they also share and angle. However, because the angle isn’t between the sides, it’s only SSA. SSA is not a valid congruency theorem, because in some cases it gives two possible solutions. In this case, we have both of them, directly connected. The question is this: what is the length of the line that is cut unevenly? That is, the one whose length equals the sum of the sides that the triangles do not share.
In today’s challenge, you will need to use trigonometry, angle theorems, and geometry.
The challenge for today is to find the area of the entire figure. We are given 3 lengths, 2 angles, 1 length equality, an angle equality, and finally, 1 set of parallel lines. Remember the units!
Today’s challenge features the concepts of geometry, angle theorems, and trigonometry. No calculator required, though.
The challenge for today is to find the area of △ABD, given that the area of △ACD is √2, that AB is 3, DC is 4, ∠BAD = 30°, and ∠ADC = 45°.